BUSI 721, Fall 2022
JGSB, Rice University
Kerry Back
\[ \bar{r}_p = w_1 \bar{r}_1 + w_2 \bar{r}_2\]
\[w_1r_1 + w_2r_2 - (w_1 \bar{r}_1 +w_2\bar{r}_2)\]
\[w_1(r_1 - \bar{r}_1) + w_2(r_2 - \bar{r}_2)\]
\[w_1(r_1 - \bar{r}_1) + w_2(r_2 - \bar{r}_2)\]
\[w_1^2 \text{var}(r_1) + w_2^2 \text{var}(r_2) + 2 w_1 w_2 \text{cov}(r_1, r_2)\]
\[\left(\frac{1}{2}\right)^2 100\text{%}^2 + \left(\frac{1}{2}\right)^2 100\text{%}^2 = 50\text{%}^2\]
Set \(\rho\) = correlation between \(r_1\) and \(r_2\)
\(\text{cov}(r_1, r_2) = \rho \times \text{std}(r_1) \times \text{std}(r_2)\)
so portfolio variance is
\[w_1^2 \text{var}(r_1) + w_2^2 \text{var}(r_2)\]
plus
\[2 w_1 w_2 \times \rho \times\text{std}(r_1)\times\text{std}(r_2)\]
Portfolio variance is lower when correlation is lower
The slope of the regression line of \(r_2\) on \(r_1\) is
\[\beta = \frac{\text{cov}(r_1, r_2)}{\text{var}(r_1)} = \rho \times \frac{\text{std}(r_2)}{\text{std(r_1)}}\]
std dev of CVX is 8.18%
std dev of AAPL is 8.50%
corr is 0.153
cov is 10.7%^2
beta is 0.16
std dev of CVX is 8.18%
std dev of XOM is 8.58%
corr is 0.889
cov is 62.4%^2
beta is 0.93